IN THE HIGHER GEOMETRY. 1 J 



Carol This property suggests a very simple and accurate 

 method of describing a conic hyperbola, and then finding its 

 centre, asymptotes, and axes ; or, any of these being given, of 

 finding the curve and the remaining paris. 



PROP. 2. Porism. A conic hyperbola being given, a point 

 may be found, such, that if from it there be drawn straight 

 lines to all the intersections of the given curve, with an 

 infinite number of parabolas, or hyperbolas, of any given 

 order whatever, lying between straight lines, of which one 

 passes through a given point, and the other may be found ; 

 the straight lines so drawn, from the point found, shall be 

 tangents to the parabolas, or hyperbolas. This is in fact two 

 propositions ; there being a construction for the case of para- 

 bolas, and another for that of hyperbolas. 



PROP. 3. Porism. If, through any point whatever of a 

 given ellipse, a straight line be drawn parallel to the con- 

 jugate axis, and cutting the ellipse in another point ; and if 

 at the first point a perpendicular be drawn to the parallel ; a 

 point may be found, such, that if from it there be drawn 

 straight lines, to the innumerable intersections of the ellipse 

 with all the parabolas of orders not given, but which may be 

 found, lying between the lines drawn at right angles to each 

 other, the lines so drawn from the point found, shall be 

 normals to the parabolas at their intersections with the 

 ellipse. 



PROP. 4. Porism. A conic hyperbola being given, if through 

 any point of it a straight line be drawn parallel to the trans- 

 verse axis, and cutting the opposite hyperbolas, a point may 

 be found, such, that if from it there be drawn straight lines, 

 to the innumerable intersections of the given curve with all 

 the hyperbolas of orders to be found, lying between straight 

 lines which may be found, the straight lines so drawn shall 

 be normals to the hyperbolas at the points of section. 



Scholium. The last two propositions give an instance of the 

 many curious and elegant analogies between the hyperbola 

 and ellipse ; failing however when, by equating the axes, we 

 change the ellipse into a circle. 



